We study the mathematical modeling and numerical simulation of the motion of red blood cells (RBC) and vesicles subject to an external incompressible flow in a microchannel. RBC and vesicles are viscoelastic bodies consisting of a deformable elastic membrane enclosing an incompressible fluid. We provide an extension of the finite element immersed boundary method by Boffi and Gastaldi (Comput Struct 81:491-501, 2003), Boffi et al. (Math Mod Meth Appl Sci 17:1479-1505, 2007), Boffi et al. (Comput Struct 85:775-783, 2007) based on a model for the membrane that additionally accounts for bending energy and also consider inflow/outflow conditions for the external fluid flow. The stability analysis requires both the approximation of the membrane by cubic splines (instead of linear splines without bending energy) and an upper bound on the inflow velocity.on the time step size is also more restrictive. We perform numerical simulations for various scenarios including the tank treading motion of vesicles in microchannels, the behavior of 'healthy' and 'sick' RBC which differ by their stiffness, and the motion of RBC through thin capillaries. The simulation results are in very good agreement with experimentally available data.
I IAbstract We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of microfluidic biochips that are used for various biom茅dical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is induced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate flnite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.Mathematics subject classification: 49K20,49M37, 65K10, 65M60, 65N30, 76N10, 76Z05, 78A70, 90C30, 92C35.
Biochips are physically and/or electronically controllable miniaturized labs. They are used for combinatorial chemical and biological analysis in environmental and medical studies. The precise positioning of the samples on the surface of the chip in picoliter to nanoliter volumes can be done either by means of external forces (active devices) or by specific geometric patterns (passive devices). The active devices which will be considered here are microfluidic biochips where the core of the technology are nanopumps featuring surface acoustic waves generated by electric pulses of high frequency. These waves propagate like a miniaturized earthquake, enter the fluid filled channels on top of the chip and cause an acoustic streaming in the fluid which provides the transport of the samples. The mathematical model represents a multiphysics problem consisting of the piezoelectric equations coupled with multiscale compressible Navier-Stokes equations that have to be treated by an appropriate homogenization. We discuss the modeling approach, present algorithmic tools for the numerical simulation and address optimal design issues. In particular, the optimal design of specific parts of the biochips leads to large-scale optimization problems. In order to reduce the computational complexity, we present a combination of domain decomposition and balanced 1 The work of the authors has been supported in part by NSF grants DMS-0511624, DMS-0707602, DMS-0810176, DMS-0811153, DMS-0914788, by AFOSR grant FA9550-09-1-0225, and by the German National Science Foundation (DFG) within the Priority Program SPP 1253 Preprint submitted to Mathematics and Computers in SimulationOctober 1, 2009 truncation model reduction which allows explicit error bounds for the error between the reduced order and the fine-scale optimization problem. It is shown that this approach gives rise to a significant reduction of the problem size while maintaining the accuracy of the approximation.
We consider an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator.
Abstract. The Immersed Boundary Method (IB) is known as a powerful technique for the numerical solution of fluid-structure interaction problems as, for instance, the motion and deformation of viscoelastic bodies immersed in an external flow. It is based on the treatment of the flow equations within an Eulerian framework and of the equations of motion of the immersed bodies with respect to a Lagrangian coordinate system including interaction equations providing the transfer between both frames. The classical IB uses finite differences, but the IBM can be set up within a finite element approach in the spatial variables as well (FE-IB). The discretization in time usually relies on the Backward Euler (BE) method for the semidiscretized flow equations and the Forward Euler (FE) method for the equations of motion of the immersed bodies. The BE/FE FE-IB is subject to a CFL-type condition, whereas the fully implicit BE/BE FE-IB is unconditionally stable. The latter one can be solved numerically by Newton-type methods whose convergence properties are dictated by an appropriate choice of the time step size, in particular, if one is faced with sudden changes in the total energy of the system. In this paper, taking advantage of the well developed affine covariant convergence theory for Newton-type methods, we study a predictor-corrector continuation strategy in time with an adaptive choice of the continuation steplength. The feasibility of the approach and its superiority to BE/FE FE-IB is illustrated by a representative numerical example.Key words. finite element immersed boundary method, fully implicit scheme, predictorcorrector continuation, red blood cells AMS subject classifications. 65H20, 65M60, 74L15, 76D05, 92C101. Introduction. A computationally attractive methodology for the numerical simulation of the motion and deformation of elastic and viscoelastic bodies in external flows is the Immersed Boundary Method (IB), which has been originally developed by Peskin [28] and further studied in [11,29,30,31,33]. The IBM uses an Eulerian coordinate system for the flow equations and Lagrangian coordinates for the boundary of the immersed bodies together with appropriate interaction equations to transform Eulerian to Lagrangian quantities and vice versa. The interaction equations feature multidimensional Dirac delta functions that have to be approximated appropriately within a finite difference approach. More recently, a variational formulation of the IBM has been provided in [8] and [9, 10] as a basis for a finite element realization referred to as the Finite Element Immersed Boundary Method (FE-IB). Both for the classical IB and the FE-IB, the most common approach with regard to discretization in time is to use the Backward Euler (BE) method for the flow equations and the Forward Euler (FE) method for the equation describing the motion and deformation of the immersed bodies which gives rise to the BE/FE IB and BE/FE FE-IB, respectively. However, these schemes typically require a CFL-type condition (cf., e.g., [9,16]). B...
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